What is the law of Syllogism?
The Law of Syllogism means that a chain of conditional statements can be combined to ignore the intermediates. Let’s say you knew that a ball went from John to Mary and then from Mary to Gloria, you could, in essence say, the ball went from John to Gloria, ignoring the intermediate steps. This can be done by chaining conditional statements. If P, then Q. And, if Q, then R. You can conclude a new conditional: if P, then R, which skips the intermediate Q. To use the law you need two things. First, you have to have two conditional statements, preferably written (or re-written if necessary) in “if-then” form. Second, the conclusion of one statement must be the logical equivalent of the hypothesis of the other statement; this is what makes this a chain. If they weren’t logical equivalents, there wouldn’t be a chain.
If John has a dollar, then he will buy candy. If someone buys candy, then they are wise. The conclusion “he will buy candy” is a logical equivalent to the hypothesis “someone buys candy”. So, you can jump to new conditional “If John has a dollar, then he is wise.”
Notice you can’t just conclude John is wise. The Law of Syllogism simply means the logic of the new conditional is VALID, it doesn’t address truth of the conclusion. This is confusing, because many books use “true” in two ways without distinguishing them. A conditional statement may be true (or valid), but that doesn’t mean its conclusion is true. A statement is true only if it NEVER is wrong. If John ever has a dollar and isn’t wise, then the new conditional isn’t true. But, once a statement is assumed to be true, then the conclusion of the statement will be found true only if the hypothesis is known to be true. So, “John is wise” would only become true if you knew that he had a dollar.
So, there are two separate things that each have their own truth value: the statement “If John has a dollar, then he is wise” and the conclusion “he is wise”.